Characterization of scaling functions in a multiresolution analysis

Authors:
P. Cifuentes, K. S. Kazarian and A. San Antolín

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1013-1023

MSC (2000):
Primary 42C15; Secondary 42C30

DOI:
https://doi.org/10.1090/S0002-9939-04-07786-X

Published electronically:
November 19, 2004

MathSciNet review:
2117202

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Abstract | References | Similar Articles | Additional Information

Abstract: We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map $A: \mathbb {R}^n\rightarrow \mathbb {R}^n$ such that $A(\mathbb {Z}^n) \subset \mathbb {Z}^n$ and all (complex) eigenvalues of $A$ have absolute value greater than $1.$ In the general case the conditions depend on the map $A.$ We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when $A$ is a diagonal matrix with all numbers in the diagonal equal to $2.$ There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.

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Additional Information

**P. Cifuentes**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Email:
patricio.cifuentes@uam.es

**K. S. Kazarian**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Email:
kazaros.kazarian@uam.es

**A. San Antolín**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Email:
angel.sanantolin@uam.es

Keywords:
Multiresolution analysis,
scaling function,
Fourier transform,
approximate continuity

Received by editor(s):
March 6, 2003

Received by editor(s) in revised form:
June 19, 2003

Published electronically:
November 19, 2004

Additional Notes:
The first two authors were partially supported by BFM2001-0189

Communicated by:
David R. Larson

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.